The problem with using digital calculations is that they are deterministic. If a result is really small, it is just rounded down to zero. So if you add a bunch of small numbers, you get zero. Even if the result should be large.
Stochastic rounding can fix this. You round each step with the probability so it's expected value is the same. Usually it will round down to 0, but sometimes it will round up to 1.
Relevant paper, using stochastic rounding. Without it the results get worse and worse before you even get to 8 bits. With stochastic rounding, there is no performance degradation. You could probably even reduce the bits even further. I think it may even be possible to get it down to 1 or 2 bits: http://arxiv.org/abs/1502.02551
Stochastic rounding can fix this. You round each step with the probability so it's expected value is the same. Usually it will round down to 0, but sometimes it will round up to 1.
Relevant paper, using stochastic rounding. Without it the results get worse and worse before you even get to 8 bits. With stochastic rounding, there is no performance degradation. You could probably even reduce the bits even further. I think it may even be possible to get it down to 1 or 2 bits: http://arxiv.org/abs/1502.02551
The relevant graph: https://i.imgur.com/cOZ4fn3.jpg